the area of triangle ABC is 38.8cm^2
Work out the length of AC
Angle c =54
base=12
(scalene triangle)
(1/2)(12)height = 38.8
height = 97/15 or 6.466..
assuming your base is BC
sin 54° = height /AC
AC = height/sin54°
take over
To find the length of side AC, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
Since the triangle is scalene, we need to find the height of the triangle first.
To find the height, we can use the sine of angle C (c):
sin(C) = height / AC
Rearranging the equation, we get:
height = AC * sin(C)
Given:
Area = 38.8 cm^2
Base (b) = 12 cm
Angle C (c) = 54 degrees
First, let's convert angle C from degrees to radians:
c = 54 * (π/180) ≈ 0.94248 radians
Now we can substitute the values into the formula to find the height:
38.8 = (1/2) * 12 * height
Divide both sides by 6:
6.467 = height * AC
Now, substitute sin(C) ≈ sin(0.94248) ≈ 0.80115:
6.467 = AC * 0.80115
To isolate AC, divide both sides by 0.80115:
AC ≈ 6.467 / 0.80115 ≈ 8.07 cm
Thus, the length of side AC is approximately 8.07 cm.
To find the length of AC in a scalene triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
First, let's find the height of the triangle by rearranging the formula:
Height = (2 * Area) / base
Given that the area of triangle ABC is 38.8 cm² and the base is 12 cm, we can calculate the height:
Height = (2 * 38.8) / 12
Height = 77.6 / 12
Height = 6.4667 cm (rounded to four decimal places)
Next, we can use the law of cosines to find the length of AC:
c² = a² + b² - 2ab * cos(C)
In this case, a and b represent the lengths of two sides of the triangle, while C represents the angle opposite side c. Given that c is the side AC with length unknown, we can rearrange the formula to solve for AC:
AC² = a² + b² - 2ab * cos(C)
Since we know the length of AB is 12 cm and angle C is 54°, we can substitute these values into the formula:
AC² = 12² + b² - 2 * 12 * b * cos(54)
To simplify this equation, we need to find the value of b. We can use the sine rule to accomplish this:
sin(B) / b = sin(C) / c
Since we know the length of BC (c) is 12 cm, and angle B is 180° - angle C - angle A, we can substitute these values into the formula:
sin(B) / b = sin(180 - 54 - 90) / 12
sin(B) / b = sin(36) / 12
sin(B) / b = 0.58779 / 12
To find out sin(B), we can use the equation sin(B) = √(1 - cos²(B)). Since cos(B) = cos(180 - C) = cos(180 - 54) = cos(126), we can substitute this value into the formula:
sin(B) = √(1 - cos²(126))
sin(B) = √(1 - (cos(126))^2)
sin(B) = √(1 - (0.5736)^2)
sin(B) = √(1 - 0.3294)
sin(B) = √(0.6706)
sin(B) = 0.8181
Now we can substitute these values back into the sine rule equation:
0.8181 / b = 0.58779 / 12
To find the value of b, we can rearrange this equation:
b = (0.8181 * 12) / 0.58779
b = 16.6951 cm (rounded to four decimal places)
Now we have the lengths of sides AB and BC, as well as the angle C. We can substitute these values into the law of cosines equation to find the length of AC:
AC² = 12² + (16.6951)² - 2 * 12 * 16.6951 * cos(54)
Since we are looking for the length of AC, we can take the square root of both sides:
AC = √ (12² + (16.6951)² - 2 * 12 * 16.6951 * cos(54))
Calculating this equation, we can find the length of AC:
AC = √ (144 + 278.7701 - 401.3416 * cos(54))
AC = √ (144 + 278.7701 - 401.3416 * 0.58779)
AC = √ (144 + 278.7701 - 235.785)
AC = √(186.9851)
AC = 13.67 cm (rounded to two decimal places)
Therefore, the length of AC in triangle ABC is approximately 13.67 cm.